Spray an 8-inch square baking dish with cooking spray and line with parchment with an inch or so of overhang on all sides.
Put the hash browns in a large bowl. Sprinkle with the cornstarch, salt and smoked paprika and toss well to coat the potatoes evenly. Add the egg and melted butter and toss well to coat. Fold in the Muenster and parsley. Press the mixture firmly into the lined baking dish and flatten the top and sides evenly. Freeze until firm, about 30 minutes.
Pour about 1/2 inch of oil in a large nonstick skillet and heat over medium-high heat. Remove the hash browns from the baking dish and cut in half with a serrated knife, then cut each half into 5 “fingers.”
The oil is ready when a shred of potato sizzles vigorously on contact. Slide half of the fingers into the oil and fry until deep golden and set on the bottom (you will know it’s time to flip when they are easy to turn), about 3 minutes. Flip and fry the second side in the same way, about 3 minutes more. Drain on a cooling rack set over a baking sheet to drain excess oil. Repeat with the remaining fingers. Season with salt and serve with ketchup on the side.